direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D5, C5⋊C6, C15⋊2C2, SmallGroup(30,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C3×D5 |
Generators and relations for C3×D5
G = < a,b,c | a3=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of C3×D5
| class | 1 | 2 | 3A | 3B | 5A | 5B | 6A | 6B | 15A | 15B | 15C | 15D | |
| size | 1 | 5 | 1 | 1 | 2 | 2 | 5 | 5 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ5 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | ζ6 | ζ65 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 6 |
| ρ6 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | ζ65 | ζ6 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 6 |
| ρ7 | 2 | 0 | 2 | 2 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ8 | 2 | 0 | 2 | 2 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ9 | 2 | 0 | -1-√-3 | -1+√-3 | -1-√5/2 | -1+√5/2 | 0 | 0 | ζ32ζ53+ζ32ζ52 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ54+ζ3ζ5 | complex faithful |
| ρ10 | 2 | 0 | -1-√-3 | -1+√-3 | -1+√5/2 | -1-√5/2 | 0 | 0 | ζ32ζ54+ζ32ζ5 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ53+ζ3ζ52 | complex faithful |
| ρ11 | 2 | 0 | -1+√-3 | -1-√-3 | -1+√5/2 | -1-√5/2 | 0 | 0 | ζ3ζ54+ζ3ζ5 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ53+ζ32ζ52 | complex faithful |
| ρ12 | 2 | 0 | -1+√-3 | -1-√-3 | -1-√5/2 | -1+√5/2 | 0 | 0 | ζ3ζ53+ζ3ζ52 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ54+ζ32ζ5 | complex faithful |
►On 15 points - transitive group 15T3
Generators in S15
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)
G:=sub<Sym(15)| (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)>;
G:=Group( (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14) );
G=PermutationGroup([[(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14)]])
G:=TransitiveGroup(15,3);
►Regular action on 30 points - transitive group 30T4
Generators in S30
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 23)(7 22)(8 21)(9 25)(10 24)(11 28)(12 27)(13 26)(14 30)(15 29)
G:=sub<Sym(30)| (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,20)(2,19)(3,18)(4,17)(5,16)(6,23)(7,22)(8,21)(9,25)(10,24)(11,28)(12,27)(13,26)(14,30)(15,29)>;
G:=Group( (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,20)(2,19)(3,18)(4,17)(5,16)(6,23)(7,22)(8,21)(9,25)(10,24)(11,28)(12,27)(13,26)(14,30)(15,29) );
G=PermutationGroup([[(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,23),(7,22),(8,21),(9,25),(10,24),(11,28),(12,27),(13,26),(14,30),(15,29)]])
G:=TransitiveGroup(30,4);
C3×D5 is a maximal subgroup of
C3⋊F5 C5⋊F7 D65⋊C3 F16⋊C2
C3×D5 is a maximal quotient of C5⋊F7 D65⋊C3 F16⋊C2
| action | f(x) | Disc(f) |
|---|---|---|
| 15T3 | x15+2x14+2x13-2x11-12x10+37x9-2x8+37x7+37x6-39x5-47x4-22x3-6x2+1 | 710·476·836·22132·23512 |
Matrix representation of C3×D5 ►in GL2(𝔽19) generated by
| 11 | 0 |
| 0 | 11 |
| 18 | 5 |
| 14 | 5 |
| 5 | 18 |
| 5 | 14 |
G:=sub<GL(2,GF(19))| [11,0,0,11],[18,14,5,5],[5,5,18,14] >;
C3×D5 in GAP, Magma, Sage, TeX
C_3\times D_5
% in TeX
G:=Group("C3xD5"); // GroupNames label
G:=SmallGroup(30,2);
// by ID
G=gap.SmallGroup(30,2);
# by ID
G:=PCGroup([3,-2,-3,-5,218]);
// Polycyclic
G:=Group<a,b,c|a^3=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C3×D5 in TeX
Character table of C3×D5 in TeX